It does not provide any Damage Threshold, but provides a bonus of 1 to Perception and can be repaired with pre-War hats. The female version is a "pillbox" hat with large black feathers in front; the male version resembles the pre-War hat, but of a different style.
The Courier can obtain this hat without permanently having the Wild Wasteland trait by selecting Wild Wasteland when you first build the Courier, then grabbing the hat, and then leaving the town boundary, where you can rebuild the Courier and take off the trait.
Non-English versions of the game translate the hat's name poorly: Spielerhut 2 in German, Sombrero de jugador 2 in Spanish, Chapeau de joueur 2 in French, and Cappello giocatore d'azzardo 2 in Italian. Each translates roughly to "Gambler Hat 2."
The following is based on unverified behind the scenes information and is speculative and may not be accurate.
The hat and its related Wild Wasteland event are a reference to an infamous scene in the 2008 film, Indiana Jones and the Kingdom of the Crystal Skull, in which the titular character survives a nuclear blast by locking himself inside of a lead refrigerator.
End of information based on unverified behind the scenes information.